Therefore ED equals GD. And DF is common, therefore the two sides ED and DF equal the two sides GD and DF, and the angle EDF equals the angle GDF, therefore the base EF equals the base GF, the triangle DEF equals the triangle DGF, and the remaining angles equal the remaining angles, namely those opposite the equal sides.

Therefore the angle DFG equals the angle DFE, and the angle DGF equals the angle DEF.

But the angle DFG equals the angle ACB, therefore the angle ACB also equals the angle DFE.

And, by hypothesis, the angle BAC also equals the angle EDF, therefore the remaining angle at B also equals the remaining angle at E. Therefore the triangle ABC is equiangular with the triangle DEF.

Therefore, if two triangles have one angle equal to one angle and the sides about the equal angles proportional, then the triangles are equiangular and have those angles equal opposite the corresponding sides.